oElectron-Phonon Coupling
in graphene

Claudio Attaccalite
Trieste 10/01/2009

Outline

DFT: ground state properties
Many-Body Perturbation Theory: excited state properties

Quasi-Particle band structure

Phonons and Electron-Phonon Coupling (EPC) in DFT (LDA and GGA)

It is possible to go beyond DFT and obtain an accurate description of the EPC

Recent Raman experiments can be reproduced completely ab-initio

Here, we show that: i) the GW approach, which provides the most accurate ab-initio treatment of electron correlation, can be used to compute the electron-phonon interaction and the phonon dispersion; ii) in graphite and graphene, DFT (LDA and GGA) underestimates by a factor 2 the slope of the phonon dispersion of the highest optical branch at the zone-boundary and the square of its electron-phonon coupling by almost 80%; iii) GW reproduces both the experimental phonon dispersion near K, the value of the EPC and the electronic band dispersion; iv) the B3LYP hybrid functional2 reproduces well the experimental phonon dispersion, but overestimates both EPC and band dispersion; v) within Hartree-Fock the graphite structure is unstable.

Density Functional Theory

Usually Exc in the local density approximation (LDA) or the general gradient approximation (GGA) successfully describes the ground state properties of solids.

The ground state energy is expressed in terms of the density and
an unknown functional Exc

Kohn-Sham eigenfunctions are obtained from

Beyond DFT:
Many-Body Perturbation Theory

Starting from the LDA Hamiltonian we construct the
Quasi-Particle Dyson equation:

: Self-Energy Operator;

: Quasi-particle energies;

... following Hedin(1965): the self-energy operator
is written as a perturbation series
of the screened Coulomb interaction

G: dressed Green Function

W: in the screened interaction

Quasi-Particles Band Structure

Comparison of GW with ARPES experiments
for graphite A. Gruenis, C.Attaccalite et al. PRL 100, 189701 (2008)

In GW the bandwidth is increased and consequently the Fermi velocity vF is enhanced

what about the phonons?

Motivations

Kohn anomalies

Phonon-mediated superconductivity

Jahn-Teller distortions

Electrical resistivity

Electron-Phonon Coupling (EPC)
determines:

The electron-phonon coupling (EPC) is one of the fundamental quantities in condensed matter. It determines phonon-dispersions and Kohn anomalies, phonon-mediated superconductivity, electrical resistivity, Jahn-Teller distortions etc.
Nowadays, density functional theory within local and semi-local approximations (DFT) is considered the standard model to compute ab-initio the electron-phonon interaction and phonon dispersions. Thus, a failure of DFT would have major consequences in a broad context. In GGA and LDA approximations,

Phonon dispersion of Graphite
(IXS measurements) and DFT calculations

M. Mohr et al. Phys. Rev. B 76, 035439 (2007)

J. Maultzusch et al. Phys. Rev. Lett. 92, 075501 (2004)

L. Wirtz and A. Rubio, Solid State Communications 131, 141 (2004)

Phonon dispersion of graphite. Lines are DFT calculations, dots and triangles are IXS measure-ments from Refs. 8,9, respectively.

Phonon dispersion close to K

In spite of the general good agreement the situation is not clear close to K

Phonon dispersion close to K

In spite of the general good agreement the situation is not clear close to K

Raman Spectroscopy of graphene

Time: 0.00 to 0.75 (min)Whenever a new material is found, Raman spectroscopy is among the first steps in order to characterize it. Graphite itself is an old and intensively studied material. However its building-block, a single-layer of graphene has only recently been transfered to a substrateDue to the high structural anisotropy it is composed of stacked layers which are only weakly coupled - few-layer graphene is a promising playground to investigate the crossover from 3D to 2D physics.Raman spectrocopy is an appropiated tool, since it probes vibrational properties and beside of that also the electronic bandstructure via the mechanism of double-resonant Raman scattering.

kK

phonon

at point, k~0 G-lineSingle-resonant
G-line

Raman G-line

K

K'

{

{

q

Raman D-line

dispersive

.... but also from Raman....

How to calculate phonons in GW?

Ideal solution: calculate total energy and its derivatives in GW

Problem: how to calculate total energy in GW

(questions of self-consistence and of numerical feasibility)

Phonon frequencies (squared) are

eigenvalues of the dynamical matrix

Time: 1.50 to 2.50 (min)Before showing results on few-layer graphene let me shortly remind you of Raman spectrosopy on graphite. A typical Raman spectrum is shown at the bottom of the slide with peaks corresponding to phonons created in the exposed graphite. Such an inelastic process is schematically depicted above: we see the electronic bandstructure with two bands crossing at the K point. We choose the one of graphene for simplicity. First, an electron is resonantly excited from the valence to the conduction band by an incoming photon. In a next step a phonon with vanishing wave vector is created, the electron will relax to a virtual state from where it will recombine, emitting a photon with a lower energy: it is this energy difference which is plotted in the horizontal axis. This so-called single-resonant process will give rise to the G line at 1582 /cm, and with a two-phonon emitted we get its overtone, the G line.

Electron-Phonon Coupling
and dynamical matrix

-bands self-energy

Electron-Phonon Coupling

Phonon dispersion without dynamical matrix of the p bands

..but using GW band structure provides a worse result

because

where

In fact the GW correction to the electronic bands alone results in a larger denominator providing a smaller phonon slope and a worse agreement with experiments.

Frozen Phonons Calculation of the EPC

The electronic Hamiltonian for the and * bands
can be written as 2x2 matrix:

a distortion of the lattice according to the G-E2G

If we diagonalize H(k,u) at the K-point

where:

!!!!!

We can get the EPC for the p bands with a frozen phonon calculation!!!

EPC and a in different approximations

To study the changes on the phonon slope we recall that Pq is the ratio of the square EPC and band energies

Thus we studied:

Hartree-Fock equilibrium structure

...the same result ...
but with another approach

Electron and phonon
renormalization of the EPC vertex

D. M. Basko et I. L. Aleiner
Phys. Rev. B 77, 0414099(R) 2008

Dirac massless fermions
+ Renormalization Group

How to model the phonon dispersion

to determine the GW phonon dispersion we assume

We assume Bq constant because it is expected to have a small dependence from q and fit it fromthe experimental measures

where

The resulting K A phonon frequency is 1192 cm1 which is our best estimation and is almost 100 cm1 smaller than in DFT.

Results: phonons around K

Phys. Rev. B 78, 081406(R) (2008) M. Lazzeri, C. Attaccalite, L. Wirtz and F. Mauri

Results: the Raman D-line dispersion

Conclusions

The GW approach can be used to calculate EPC

In graphene and graphite DFT(LDA and GGA) underestimates the phonon dispersion of the highest optical branch at the zone-boundaries

B3LYP gives a results similar to GW but overestimates the EPC

It is possible to reproduce completely ab-initio the Raman D-line shift

Acknowledgment

http://www.pwscf.org

http://www.yambo-code.org

2) My collaborators
M. Lazzeri, L. Wirtz, A. Rubio and F. Mauri

Codes I used:

1) The support from:

http://www.crystal.unito.it

The case of Doped Graphene

Therefore the effective interaction felt by the electrons starts to be weaker due
the stronger screening of the Coulomb potential.

With doping graphene evolves from a semi-metal to a real metal.

http://arxiv.org/abs/0808.0786 C. Attaccalite et al.

Solid State Commun. 143, 58 (2007) M. Polini et al.

Electron-phonon Coupling at K

LDA results is recovered in doping graphene

WORK IN PROGRESS . . .

Tuning the B3LYP

The B3LYP hybrid-functional has the from:

B3LYP consist of a mixture of Vosko-Wilk-Nusair and LYP correlation part Ec
and a mixture of LDA/Becke exchange with Hartree-Fock exchange

The parameter A controls the admixture
of HF exchange in the standard B3LYP is 20%

It is possible to reproduce GW results
tuning the non-local exchange in B3LYP !!!!!

Many-Body perturbation Theory 2

Approximations for G and W (Hybertsen and Louie, 1986): Random phase approximation (RPA) for the dielectric function. General plasmon-pole model for dynamical screening.

we use the LDA results as starting point

and so the Dyson equation becomes

Raman spectroscopy of graphene

kK

phonon

at point, k~0 G-lineSingle-resonant
G-line

Ref.: S. Reich, C. Thomsen, J. Maultzsch, Carbon Nanotubes, Wiley-VCH (2004)

D-line

graphite2.33 eVDGDG

TO mode between K and M

dispersive

Time: 1.50 to 2.50 (min)Before showing results on few-layer graphene let me shortly remind you of Raman spectrosopy on graphite. A typical Raman spectrum is shown at the bottom of the slide with peaks corresponding to phonons created in the exposed graphite. Such an inelastic process is schematically depicted above: we see the electronic bandstructure with two bands crossing at the K point. We choose the one of graphene for simplicity. First, an electron is resonantly excited from the valence to the conduction band by an incoming photon. In a next step a phonon with vanishing wave vector is created, the electron will relax to a virtual state from where it will recombine, emitting a photon with a lower energy: it is this energy difference which is plotted in the horizontal axis. This so-called single-resonant process will give rise to the G line at 1582 /cm, and with a two-phonon emitted we get its overtone, the G line.

Electron-Hole Coupling

A(%)M gapaK

12%176.965.54731.93

13%185.505.66232.99

14%194.395.69534.13

15%203.655.76935.30

20%256.036.14041.70

GW1934.8939.5

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